KnowledgeBoat Logo
|

Mathematics

Prove the following identity:

(1+tan⁡2A)+(1+cot⁡2A)=(1sin⁡2A−sin⁡4A)(1 + \tan^2 A) + (1 + \cot^2 A) = \Big(\dfrac{1}{\sin^2 A - \sin^4 A}\Big)

Trigonometric Identities

3 Likes

Answer

Solving L.H.S. of the equation :

⇒(1+tan⁡2A)+(1+cot⁡2A)⇒sec⁡2A+cosec⁡2A⇒1cos⁡2A+1sin⁡2A⇒sin⁡2A+cos⁡2Acos⁡2Asin⁡2A⇒1cos⁡2Asin⁡2A⇒1(1−sin⁡2A)sin⁡2A⇒1sin⁡2A−sin⁡4A.\Rightarrow (1 + \tan^2 A) + (1 + \cot^2 A) \\[1em] \Rightarrow \sec^2 A + \cosec^2 A \\[1em] \Rightarrow \dfrac{1}{\cos^2 A} + \dfrac{1}{\sin^2 A} \\[1em] \Rightarrow \dfrac{\sin^2 A + \cos^2 A}{\cos^2 A \sin^2 A} \\[1em] \Rightarrow \dfrac{1}{\cos^2 A \sin^2 A} \\[1em] \Rightarrow \dfrac{1}{(1 - \sin^2 A) \sin^2 A} \\[1em] \Rightarrow \dfrac{1}{\sin^2 A - \sin^4 A} .

Since, L.H.S. = R.H.S.

Hence, proved that (1+tan⁡2A)+(1+cot⁡2A)=(1sin⁡2A−sin⁡4A)(1 + \tan^2 A) + (1 + \cot^2 A) = \Big(\dfrac{1}{\sin^2 A - \sin^4 A}\Big).

Answered By

1 Like

Related Questions